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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

邮发代号 80-964

2019 Impact Factor: 1.03

Frontiers of Mathematics in China  2016, Vol. 11 Issue (5): 1189-1206   https://doi.org/10.1007/s11464-016-0582-3
  本期目录
A glance at three-dimensional Alexandrov spaces
Fernando GALAZ-GARCÍA()
Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany
 全文: PDF(227 KB)  
Abstract

We discuss the topology and geometry of closed Alexandrov spaces of dimension three.

Key wordsAlexandrov space    group action    3-manifold
收稿日期: 2016-05-23      出版日期: 2016-09-23
Corresponding Author(s): Fernando GALAZ-GARCÍA   
 引用本文:   
. [J]. Frontiers of Mathematics in China, 2016, 11(5): 1189-1206.
Fernando GALAZ-GARCÍA. A glance at three-dimensional Alexandrov spaces. Front. Math. China, 2016, 11(5): 1189-1206.
 链接本文:  
https://academic.hep.com.cn/fmc/CN/10.1007/s11464-016-0582-3
https://academic.hep.com.cn/fmc/CN/Y2016/V11/I5/1189
1 Berestovskiĭ V N. Homogeneous manifolds with an intrinsic metric II. Sibirsk Mat Zh, 1989, 30(2): 14–28 (in Russian); Sib Math J, 1989, 30(2): 180–191
2 Bredon G E. Introduction to Compact Transformation Groups. New York: Acad Press, 1972
3 Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. Grad Stud Math, Vol 33. Providence: Amer Math Soc, 2001
4 Burago Y, Gromov M, Perel’man G. A. D. Alexandrov spaces with curvatures bounded below. Uspekhi Mat Nauk, 1992, 47(2): 3–51(in Russian); Russian Math Surveys, 1992, 47(2): 1–58
https://doi.org/10.1070/RM1992v047n02ABEH000877
5 Cannon J W. Shrinking cell-like decompositions of manifolds. Codimension three. Ann of Math (2), 1979, 110(1): 83–112
6 Dantzig D van, Waerden B Lvan der. Über metrisch homogene r¨aume. Abh Math Sem Univ Hamburg, 1928, 6(1): 367–376
https://doi.org/10.1007/BF02940622
7 Deng Q, Galaz-Garćıa F, Guijarro L, Munn M. Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature. Preprint, 2016, arXiv: 1602.07724
8 Dinkelbach J, Leeb B. Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom Topol, 2009, 13(2): 1129–1173
https://doi.org/10.2140/gt.2009.13.1129
9 Edwards R D. The topology of manifolds and cell-like maps. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Helsinki: Acad Sci Fennica, 1980, 111–127
10 Edwards R D. Suspensions of homology spheres. Preprint, 2006, arXiv: math/0610573
11 Fukaya K, Yamaguchi T. Isometry groups of singular spaces. Math Z, 1994, 216(1): 31–44
https://doi.org/10.1007/BF02572307
12 Galaz-Garćıa F, Guijarro L. Isometry groups of Alexandrov spaces. Bull Lond Math Soc, 2013, 45(3): 567–579
https://doi.org/10.1112/blms/bds101
13 Galaz-Garćıa F, Guijarro L. On three-dimensional Alexandrov spaces. Int Math Res Not IMRN, 2015, (14): 5560–5576
14 Galaz-Garćıa F, Searle C. Cohomogeneity one Alexandrov spaces. Transform Groups, 2011, 16(1): 91–107
https://doi.org/10.1007/s00031-011-9122-0
15 Galaz-Garćıa F, Zarei M. Cohomogeneity one topological manifolds revisited. Preprint, 2015, arXiv: 1503.09068
16 Grove K. Geometry of, and via, symmetries. In: Conformal, Riemannian and Lagrangian Geometry (Knoxville, TN, 2000). Univ Lecture Ser, Vol 27. Providence: Amer Math Soc, 2002, 31–53
https://doi.org/10.1090/ulect/027/02
17 Grove K. Developments around positive sectional curvature. In: Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry. Surv Differ Geom, Vol 13. Somerville: Int Press, 2009, 117–133
18 Grove K, Petersen P. Manifolds near the boundary of existence. J Differential Geom, 1991, 33(2): 379–394
19 Grove K, Wilking B. A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. Geom Topol, 2014, 18(5): 3091–3110
https://doi.org/10.2140/gt.2014.18.3091
20 Harvey J, Searle C. Orientation and symmetries of Alexandrov spaces with applications in positive curvature. Preprint, 2013, arXiv: 1209.1366
21 Hempel J. 3-manifolds. Providence: AMS Chelsea Publishing, 2004
https://doi.org/10.1090/chel/349
22 Hirsch M W, Smale S. On involutions of the 3-sphere. Amer J Math, 1959, 81: 893–900
https://doi.org/10.2307/2372994
23 Hoelscher C A. Classification of cohomogeneity one manifolds in low dimensions. Pacific J Math, 2010, 246(1): 129–185
https://doi.org/10.2140/pjm.2010.246.129
24 Hsiang W-Y. Lie transformation groups and differential geometry. In: Differential Geometry and Differential Equations (Shanghai, 1985). Lecture Notes in Math, Vol 1255. Berlin: Springer, 1987, 34–52
https://doi.org/10.1007/BFb0077679
25 Kapovitch V. Perelman’s stability theorem. In: Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 103–136
26 Kobayashi S. Transformation Groups in Differential Geometry. Classics Math. Berlin: Springer-Verlag, 1995
27 Kwun K W, Tollefson J L. PL involutions of S1× S1 × S1.Trans Amer Math Soc, 1975, 203: 97–106
28 Livesay G R. Involutions with two fixed points on the three-sphere. Ann of Math (2), 1963, 78: 582–593
29 Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann of Math (2), 2009, 169(3): 903–991
30 Luft E, Sjerve D. Involutions with isolated fixed points on orientable three-dimensional flat space forms. Trans Amer Math Soc, 1984, 285(1): 305–336
31 Mitsuishi A, Yamaguchi T. Collapsing three-dimensional closed Alexandrov spaces with a lower curvature bound. Trans Amer Math Soc, 2015, 367(4): 2339–2410
https://doi.org/10.1090/S0002-9947-2014-06091-1
32 Mostert P S. On a compact Lie group acting on a manifold. Ann of Math (2), 1957, 65: 447–455; Errata, “On a compact Lie group acting on a manifold”. Ann of Math (2), 1957, 66: 589
33 Myers S B, Steenrod N E. The group of isometries of a Riemannian manifold. Ann of Math (2), 1939, 40(2): 400–416
34 Neumann W D. 3-Dimensional G-manifolds with 2-dimensional orbits. In: Mostert P S, ed. Proceedings of the Conference on Transformation Groups. Berlin: Springer- Verlag, 1968, 220–222
https://doi.org/10.1007/978-3-642-46141-5_16
35 Núñez-Zimbrón J. Closed three-dimensional Alexandrov spaces with isometric circle actions. Tohoku Math J (to appear), arXiv: 1312.0540
36 Orlik P. Seifert Manifolds. Lecture Notes in Math, Vol 291. Berlin: Springer-Verlag, 1972
https://doi.org/10.1007/BFb0060329
37 Orlik P, Raymond F. Actions of SO(2) on 3-manifolds. In: Mostert P S, ed. Proceedings of the Conference on Transformation Groups. Berlin: Springer-Verlag, 1968, 297–318
https://doi.org/10.1007/978-3-642-46141-5_22
38 Otsu Y, Shioya T. The Riemannian structure of Alexandrov spaces. J Differential Geom, 1994, 39(3): 629–658
39 Parker J. 4-Dimensional G-manifolds with 3-dimensional orbits. Pacific J Math, 1986, 129(1): 187–204
https://doi.org/10.2140/pjm.1986.125.187
40 Perelman G. Alexandrov spaces with curvatures bounded from below, II. Preprint, 1991
41 Perelman G. Elements of Morse theory on Aleksandrov spaces. St Petersburg Math J, 1994, 5: 205–213
42 Perel’man G Ya, Petrunin A M. Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem. Algebra i Analiz, 1993, 5(1): 242–256 (Russian); St Petersburg Math J, 1994, 5(1): 215–227
43 Petrunin A. Applications of quasigeodesics and gradient curves. In: Comparison Geometry (Berkeley, CA, 1993-94). Math Sci Res Inst Publ, Vol 30. Cambridge: Cambridge Univ Press, 1997, 203–219
44 Petrunin A. Semiconcave functions in Alexandrov’s geometry. In: Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 137–201
45 Plaut C. Metric spaces of curvature _ k. In: Handbook of Geometric Topology. Amsterdam: North-Holland, 2002, 819–898
46 Porti J. Geometrization of three manifolds and Perelman’s proof. Rev R Acad Cienc Exactas F´ıs Nat Ser A Mat RACSAM, 2008, 102(1): 101–125
47 Raymond F. Classification of the actions of the circle on 3-manifolds. Trans Amer Math Soc, 1968, 131: 51–78
https://doi.org/10.2307/1994680
48 Scott P. The geometries of 3-manifolds. Bull Lond Math Soc, 1983, 15(5): 401–487
https://doi.org/10.1112/blms/15.5.401
49 Shiohama K. An Introduction to the Geometry of Alexandrov Spaces. Lecture Notes Series, 8. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993
50 Shioya T, Yamaguchi T. Collapsing three-manifolds under a lower curvature bound. J Differential Geom, 2000, 56(1): 1–66
51 Shioya T, Yamaguchi T. Volume collapsed three-manifolds with a lower curvature bound. Math Ann, 2005, 333(1): 131–155
https://doi.org/10.1007/s00208-005-0667-x
52 Sturm K-T. On the geometry of metric measure spaces. I. Acta Math, 2006, 196(1): 65–131
https://doi.org/10.1007/s11511-006-0002-8
53 Sturm K-T. On the geometry of metric measure spaces. II. Acta Math, 2006, 196(1): 133–177
https://doi.org/10.1007/s11511-006-0003-7
54 Wilking B. Nonnegatively and positively curved manifolds. In: Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 25–62
55 Ziller W. Riemannian manifolds with positive sectional curvature. In: Herrera R, Hernández-Lamoneda L, eds. Geometry of Manifolds with Non-negative Sectional Curvature. Lecture Notes in Math, Vol 2110. Cham: Springer, 2014<Date>,</Date> 1–19
https://doi.org/10.1007/978-3-319-06373-7_1
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